Mage's Rhizome

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====== Conditional evolution equation ======

We follow the derivation in [(:cite:cwf:norsen2015)] (§3) wich seems easier to understand than the equivalent treatment in [(:cite:cwf:oriols2007)].

Take the [[conditional wave function]] $\psi(t,x) = \Psi(t,x,Y(t))$ and put it into the usual Schrödinger equation with arbitrary potential $V(t,x,y)$. Note that the time derivative will also act on $Y(t)$ invoking the chain rule of differential calculus. We write $\psi' = \nabla_y \Psi|_{y=Y(t)}$ and $\psi'' = \Delta_y\Psi|_{y=Y(t)}$ like in [(:cite:cwf:norsen2015)] (25-26) to shorten notation.

\begin{equation}
\begin{aligned}
\mathrm{i}\hbar \partial_t \psi &= \mathrm{i}\hbar\partial_t\Psi|_{y=Y(t)} + \mathrm{i}\hbar \dot{Y} \cdot \nabla_y \Psi|_{y=Y(t)} \\
&= -\frac{\hbar^2}{2m}\Delta_x\psi -\frac{\hbar^2}{2m}\Delta_y\Psi|_{y=Y(t)} + V(t,x,Y(t))\psi  + \mathrm{i}\hbar \dot{Y} \cdot \nabla_y \Psi|_{y=Y(t)} \\
&= -\frac{\hbar^2}{2m}\Delta_x\psi + V(t,x,Y(t))\psi + \mathrm{i}\hbar \dot{Y} \cdot \psi' -\frac{\hbar^2}{2m}\psi''
\end{aligned}
\end{equation}

We see that two terms on the right resemble the usual one-particle Schrödinger equation and two other terms are added due to apparent entanglement effects. By defining two auxiliary potentials $A$ and $B$ we can rewrite the equation above into a Schrödinger equation with an effective potential steering the [[conditional wave function]].

\begin{align}
A(t,x) &= \mathrm{i}\hbar \dot{Y}(t) \cdot \frac{\psi'(t,x)}{\psi(t,x)} \\
B(t,x) &= -\frac{\hbar^2}{2m} \frac{\psi''(t,x)}{\psi(t,x)}
\end{align}

Overall we get the following conditional evolution equation
\begin{equation}
\mathrm{i}\hbar \partial_t \psi = -\frac{\hbar^2}{2m}\Delta_x\psi + (V(t,x,Y(t)) + A(t,x) + B(t,x))\psi
\end{equation}
where the contributions $A$ and $B$ depend on $\psi$ again making the whole equation non-linear, and further may also be complex thus defining a non-unitary evolution.

The $A$ and $B$ still include, through $\psi'$ and $\psi''$, the full wave function $\Psi$ on the whole configuration space, so this is not yet an autonomous evolution equation for the conditional wave function. But there is actually a way how to eliminate $\Psi$ and give separate evolution equations for the potentials if only in the form of an infinite [[potential network]] or directly with a [[coupled system]] of $\psi,\psi',\psi''$ etc.

If the same treatment is //vice versa// employed for the subsystem $Y$, taking the Bohmian trajectory $x = X(t)$ to accomodate for influences of the subsystem $X$, the resulting approximation scheme is called "Bohmian Double Semi-Quantum (BDSQ)" approximation in [(:cite:cwf:norsen2016)].